Question: $\overline{AB} = 52$ $\overline{BC} = {?}$ $A$ $C$ $B$ $52$ $?$ $ \sin( \angle ABC ) = \dfrac{12}{13}, \cos( \angle ABC ) = \dfrac{5}{13}, \tan( \angle ABC ) = \dfrac{12}{5}$
Solution: $\overline{AB}$ is the hypotenuse $\overline{BC}$ is adjacent to $\angle ABC$ SOH CAH TOA We know the hypotenuse and need to solve for the adjacent side so we can use the cos function (CAH) $ \cos( \angle ABC ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\overline{BC}}{\overline{AB}}= \frac{\overline{BC}}{52} $ Since we have already been given $\cos( \angle ABC )$ , we can set up a proportion to find $\overline{BC}$ $ \cos( \angle ABC ) = \dfrac{5}{13} = \frac{\overline{BC}}{52}$ Simplify. $\overline{BC} = 20$